class 12

Missing

JEE Advanced

A circuit is connected as shown in the figure with the swith S open. When the swith is closed, the total amount of charge that flows from Y to X is

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For a non-zero complex number $z$, let $arg(z)$denote theprincipal argument with $π<arg(z)≤π$Then, whichof the following statement(s) is (are) FALSE?$arg(−1,−i)=4π ,$where $i=−1 $(b) The function $f:R→(−π,π],$defined by $f(t)=arg(−1+it)$for all $t∈R$, iscontinuous at all points of $R$, where $i=−1 $(c) For any two non-zero complex numbers $z_{1}$and $z_{2}$, $arg(z_{2}z_{1} )−arg(z_{1})+arg(z_{2})$is an integer multiple of $2π$(d) For any three given distinct complex numbers $z_{1}$, $z_{2}$and $z_{3}$, the locus of the point $z$satisfying the condition $arg((z−z_{3})(z_{2}−z_{1})(z−z_{1})(z_{2}−z_{3}) )=π$, lies on a straight line

Let $y(x)$ be a solution of the differential equation $(1+e_{x})y_{prime}+ye_{x}=1.$ If $y(0)=2$ , then which of the following statements is (are) true? (a)$y(−4)=0$ (b)$y(−2)=0$ (c)$y(x)$ has a critical point in the interval $(−1,0)$ (d)$y(x)$ has no critical point in the interval$(−1,0)$

For any positive integer $n$, define $f_{n}:(0,∞)→R$as $f_{n}(x)=j=1∑n tan_{−1}(1+(x+j)(x+j−1)1 )$for all $x∈(0,∞)$.Here, the inverse trigonometric function $tan_{−1}x$assumes values in $(−2π ,2π )˙$Then, which of the following statement(s) is (are) TRUE?$j=1∑5 tan_{2}(f_{j}(0))=55$(b) $j=1∑10 (1+fj_{′}(0))sec_{2}(f_{j}(0))=10$(c) For any fixed positive integer $n$, $(lim)_{x→∞}tan(f_{n}(x))=n1 $(d) For any fixed positive integer $n$, $(lim)_{x→∞}sec_{2}(f_{n}(x))=1$

Suppose that $p ,q andr$ are three non-coplanar vectors in $R_{3}$. Let the components of a vector $s$ along $p ,q andr$ be 4, 3 and 5, respectively. If the components of this vector $s$ along $(−p +q +r),(p −q +r)and(−p −q +r)$ are x, y and z, respectively, then the value of $2x+y +z$ is

Word of length 10 are formed using the letters A,B,C,D,E,F,G,H,I,J. Let $x$be the number of such words where no letter is repeated; and let $y$be the number of such words where exactly one letter is repeated twice and no other letter is repeated. The, $9xy =$

Let $ω$be a complex cube root of unity with $ω=1andP=[p_{ij}]$be a $n×n$matrix withe $p_{ij}=ω_{i+j}˙$Then $p_{2}=O,whe∩=$a.$57$b. $55$c. $58$d. $56$

Three randomly chosen nonnegative integers $x,yandz$are found to satisfy the equation $x+y+z=10.$Then the probability that $z$is even, is:$125 $ (b) $21 $ (c) $116 $ (d) $5536 $

A curve passes through the point $(1,6π )$ . Let the slope of the curve at each point $(x,y)$ be $xy +sec(xy ),x>0.$ Then the equation of the curve is